Vector Addition and Subtraction
CSEC Mathematics: Combining Vectors
Essential Understanding: Vectors can be added or subtracted to find resultant vectors. This is essential for solving problems in navigation, physics, and geometry. Understanding the Triangle Law and Parallelogram Law will help you visualize and calculate vector operations.
Adding Vectors Algebraically
When adding vectors in column form, simply add the corresponding components:
Vector Addition Formula
Add x-components together, add y-components together
Worked Example 1: Basic Addition
Calculate: \(\begin{pmatrix} 3 \\ 5 \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \end{pmatrix}\)
Answer: \(\begin{pmatrix} 3 \\ 5 \end{pmatrix} + \begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ 4 \end{pmatrix}\)
Subtracting Vectors Algebraically
When subtracting vectors, subtract the corresponding components:
Vector Subtraction Formula
Subtract x-components, subtract y-components
Worked Example 2: Vector Subtraction
Calculate: \(\begin{pmatrix} 7 \\ 2 \end{pmatrix} - \begin{pmatrix} 4 \\ 6 \end{pmatrix}\)
Answer: \(\begin{pmatrix} 7 \\ 2 \end{pmatrix} - \begin{pmatrix} 4 \\ 6 \end{pmatrix} = \begin{pmatrix} 3 \\ -4 \end{pmatrix}\)
Remember: Subtraction as Addition
\(\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})\)
Subtracting a vector is the same as adding its negative (inverse) vector.
Scalar Multiplication of Vectors
Multiplying a vector by a scalar (number) multiplies each component:
Scalar Multiplication
Worked Example 3: Scalar Multiplication
Calculate: \(3\begin{pmatrix} 2 \\ -4 \end{pmatrix}\)
Solution: \(3\begin{pmatrix} 2 \\ -4 \end{pmatrix} = \begin{pmatrix} 3 \times 2 \\ 3 \times (-4) \end{pmatrix} = \begin{pmatrix} 6 \\ -12 \end{pmatrix}\)
Geometric Interpretation: Triangle Law
The Triangle Law of Vector Addition
If two vectors are represented by two sides of a triangle taken in order, then their resultant (sum) is represented by the third side taken in the opposite direction.
To add a and b: Place the tail of b at the head of a. The resultant goes from the tail of a to the head of b.
Geometric Interpretation: Parallelogram Law
The Parallelogram Law of Vector Addition
If two vectors are represented by two adjacent sides of a parallelogram, then their resultant is represented by the diagonal passing through their common point.
Both vectors start from the same point. The diagonal of the parallelogram gives the resultant.
Interactive Vector Addition Lab
Vector Addition Calculator
Enter two vectors and see their addition visualized using the Triangle Law!
\(\begin{pmatrix} 3 \\ 2 \end{pmatrix}\)
\(\begin{pmatrix} 2 \\ 4 \end{pmatrix}\)
\(\begin{pmatrix} 5 \\ 6 \end{pmatrix}\)
Vector Subtraction Geometrically
Understanding a - b
To subtract vector b from a, we add the negative of b:
\(\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})\)
Position Vectors
A position vector describes the position of a point relative to the origin. If point P has coordinates (x, y), its position vector is:
Worked Example 4: Finding a Vector Between Two Points
Given: Point A(2, 3) and Point B(7, 1)
Find: \(\overrightarrow{AB}\)
Answer: \(\overrightarrow{AB} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\)
Important Formula
To find the vector from point A to point B:
\(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}\)
Or simply: subtract coordinates of A from coordinates of B
CSEC Practice Questions
Test Your Understanding
\(2\mathbf{a} = 2\begin{pmatrix} 2 \\ 5 \end{pmatrix} = \begin{pmatrix} 4 \\ 10 \end{pmatrix}\)
\(2\mathbf{a} + \mathbf{b} = \begin{pmatrix} 4 \\ 10 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 3 \\ 13 \end{pmatrix}\)
CSEC Examination Tips
- Be careful with signs! Double-check when subtracting negative numbers.
- Draw diagrams when dealing with position vectors - it helps visualize the problem.
- For \(\overrightarrow{AB}\): Remember it's always "End minus Start" (B minus A).
- Common CSEC question: Given position vectors, find the vector between two points and its magnitude.